Deformation quantization modules

Abstract

We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class commutes with composition of kernels, a kind of Riemann-Roch theorem in the non-commutative setting. Finally we study holonomic modules on complex symplectic manifolds and we prove in particular a constructibility theorem.

This paper develops the results of Deformation quantization modules I (arXiv:0802.1245) and II (arXiv:0809.4309), and also contains new results and new remarks. It will appear in Asterisque, Soc. Math. France (2012)

References in corpus (3)

• The relative Riemann-Roch theorem from Hochschild homology
• Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics
• Regular holonomic D[[h]]-modules

Cited by in corpus (37)

• Quantizations of conical symplectic resolutions I: local and global structure
• Wrapped microlocal sheaves on pairs of pants
• Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Twisted Deformation Quantization of Algebraic Varieties
• A categorical action on quantized quiver varieties
• Duality and Tilting for Commutative DG Rings
• The finiteness conjecture for skein modules
• Regular holonomic D[[h]]-modules
• Quantisation of derived Lagrangians
• The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles
• Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles
• Bivariance, Grothendieck duality and Hochschild homology, II: the fundamental class of a flat scheme-map
• Remarks on derived complete modules and complexes
• Microlocal Euler classes and Hochschild homology
• Geometry and categorification
• On quantization of complex symplectic manifolds
• Morita classes of microdifferential algebroids
• The curvature problem for formal and infinitesimal deformations
• On a conjecture of Kashiwara relating Chern and Euler classes of O-modules
• The ADO Invariants are a q-Holonomic Family
• Hochschild Lefschetz Class for $D$-modules
• NC-smooth algebroid thickenings for families of vector bundles and quiver representations
• Formality of differential graded algebras and complex Lagrangian submanifolds
• Todd class via homotopy perturbation theory
• A variant of the Mukai pairing via deformation quantization
• Derived equivalence for quantum symplectic resolutions
• Microdifferential systems and the codimension-three conjecture
• A note on quantization of complex symplectic manifolds
• The Lefschetz-Lunts formula for deformation quantization modules
• A Riemann-Roch formula for traces of algebraic differential operators
• Reconstruction of a variety from $\mathcal{O}[[\hslash ]]$-modules
• On the codimension-three conjecture
• Regularity for elliptic pairs over C[[h]]
• The $\hat{A}$ genus as a projective volume form on the derived loop space
• DG-resolutions of NC-smooth thickenings and NC-Fourier-Mukai transforms
• Deformations of Affine Varieties and the Deligne Crossed Groupoid